Strong conciseness of coprime and anti-coprime commutators

TL;DR

This paper investigates the structure of profinite groups by analyzing the cardinality of specific types of commutators, establishing conditions under which the groups are finite-by-pronilpotent or have finite commutator subgroup.

Contribution

It proves that small sets of coprime or anti-coprime commutators imply significant structural properties in profinite groups.

Findings

Profinite groups with less than continuum coprime commutators are finite-by-pronilpotent.

Groups with fewer than continuum anti-coprime commutators have finite commutator subgroup.

The results connect commutator set size to profound group-theoretic properties.

Abstract

A coprime commutator in a profinite group $G$ is an element of the form $[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the same primes. In the present paper we establish that a profinite group $G$ is finite-by-pronilpotent if the cardinality of the set of coprime commutators in $G$ is less than $2^{\aleph_0}$. Moreover, a profinite group $G$ has finite commutator subgroup $G'$ if the cardinality of the set of anti-coprime commutators in $G$ is less than $2^{\aleph_0}$.